The Significance of Digital GeneExpression ProfilesSte´phane Audic and Jean-Michel Claverie1Laboratory of Structural and Genetic Information,Centre National de la Recherche Scientifique–E.P.91,Marseille13402,FranceGenes differentially expressed in different tissues,during development,or during specific pathologies are of foremost interest to both basic and pharmaceutical research.‘‘Transcript profiles’’or‘‘digital Northerns’’are generated routinely by partially sequencing thousands of randomly selected clones from relevant cDNA libraries.Differentially expressed genes can then be detected from variations in the counts of their cognate sequence tags.Here we present the first systematic study on the influence of random fluctuations and sampling size on the reliability of this kind of data.We establish a rigorous significance test and demonstrate its use on publicly available transcript profiles.The theory links the threshold of selection of putatively regulated genes (e.g.,the number of pharmaceutical leads)to the fraction of false positive clones one is willing to risk.Our results delineate more precisely and extend the limits within which digital Northern data can be used.Very large-scale,single-pass partial sequencing of cDNA clones from a large number of libraries has led to the identification of∼50,000human genes(Ad-ams et al.1995;Aaronson et al.1996;Hillier et al. 1996).However,a precise function or a complete transcript sequence are known for<5000of these (Adams et al.1995;Boguski and Schuler1995).In the absence of functional clues for most of the newly identified genes,evidence of differential ex-pression is the most important criteria to prioritize the exploitation of anonymous sequence data in both basic and pharmaceutical(Nowak1995;Ad-ams1996;Bains1996;Editorial1996)research.For example,the study of expression profiles in various tumors is central to the new Cancer Genome Anatomy project(Kuska1996;O’Brien1997).In contrast to functional assays,the quantitative analysis of gene expression level lends itself to large-scale implementation.Two main approaches have been proposed(1)‘‘analog’’methods based on hy-bridization to arrayed cDNA libraries(Lennon and Lehrach1991;Gress et al.1992;Nguyen et al.1995; Schena et al.1995;Zhao et al.1995)or oligonucleo-tide‘‘chips’’(Fodor et al.1991;Southern et al.1992; Guo et al.1994;Matson et al.1995);and(2)‘‘digi-tal’’methods,based on the generation of sequence tags.This paper focuses on the latter.The sequence tag-based method(Okubo et al.1992;Matsubara and Okubo1994)consists of generating a large number(thousands)of expressed sequence tags (ESTs)(Adams et al.1991;Wilcox et al.1991;Adams et al.1992;Khan et al.1992)from3Ј-directed re-gional non-normalized cDNA libraries.Recently, Velculescu et al.(1995)have introduced the serial analysis of gene expression(SAGE).Although tags are100–300nucleotides in length in the original EST approach,the SAGE method only requires nine nucleotides,therefore allowing a larger throughput. In both protocols,the number of tags is reported to be proportional to the abundance of cognate tran-scripts in the tissue or cell type used to make the cDNA library.The variation in the relative fre-quency of those tags,stored in computer databases, is then used to point out the differential expression of the corresponding genes:This is the concept of a ‘‘digital Northern’’comparison.In the absence of a sound theoretical framework,the validity of the method has only been verified for a handful of genes in the context of two cellular differentiation systems(Lee et al.1995;Okubo et al.1995)induc-ible in vitro.Yet,with a total number of human genes of∼80,000or more,it is not intuitive that sequencing a mere few thousand tags(a typical ex-periment)from highly redundant non-normalized cDNA libraries can produce a useful picture,or real-istic‘‘transcript profile,’’of a given tissue,develop-ment stage,or cell type.What variations in tag numbers allow for a reliable inference about differ-ential expression?How many tags should be gener-ated?Here we present the statistical framework re-quired to answer those questions and analyze tran-script profiles in a quantitative manner.1Corresponding author.E-MAIL jmc@rs-mrs.fr;FAX33491164549.RESEARCH 986RESULTSIn Methods we establish the probability distribution governing the occurrence of the same rare event in duplicate experiments.This probability distribution is a general result applicable to a wide variety of experimental situations,although this paper fo-cuses on its use to analyze digital gene expression patterns.The main and only mathematical assump-tion behind the derivation is that the observed events are rare and part of a large population of possible outcomes(the distribution of which is not specified).In the context of a digital Northern,one event is the observation of a given cDNA sequence tag,and the experiment consists of the random picking and partial sequencing of a number N of cDNA clones.Given the usual complexity(i.e.,the number of different genes expressed)of cDNA li-braries,observing a given cDNA qualifies as a rare event,as the abundance of most individual mes-sages is of the order of a few percents or less. Random Fluctuation vs.Significant Change in Tag Number:When to Infer Differential ExpressionLet us randomly pick N=1000clones from a cDNA library and generate the corresponding sequence tags;a given message(e.g.,interleukin-2)will be picked x(e.g.,two)times,with x in a typical(0–10) range.If we now redo this experiment,that is,again pick1000clones and generate the tags,the same message will now be picked y(e.g.,3)times.If the experiments have been duplicated correctly and the clones selected at random,we expect x and y to be close,albeit often different because of random fluc-tuations.In the Methods section,we show that the expected probability of observing y occurrences of a clone already observed x times is given by the simple formula:p(y|x)=͑x+y͒!x!y!2͑x+y+1͒(1)Equation1can be used to compute a confi-dence interval[y min,y max]⑀within which we expect to find y with a given probability,noted1–2⑀,where 2⑀is the significance level.For⑀small(e.g.,2.5%or less),y values falling outside the[y min,y max]⑀inter-val correspond to p(y|x)<<1,therefore pointing out very unlikely random fluctuations between the two experiments.The confidence intervals for the usual1%and5%significance levels are given in Table1.The same confidence intervals listed in Table1 can in fact be used to analyze the results of sampling N clones from two different libraries.Provided all experimental factors are well replicated,significant discrepancies between x(from one library)and y (from the other)will now characterize differentially expressed genes,for example,the relative abun-dance of which is unlikely to be the same in the two libraries.Simply reading Table1,we see that varia-tions in counts such as7→0,or2→12are signifi-cant(P<0.01)evidence of regulated gene expres-sion,whereas variations such as3→0or8→16are not(P>0.05).However,we do not advocate the use of rigid significance thresholds to analyze digital transcript profiles,as discussed below.Influence of the Sampling SizeSurprisingly at first,p(y|x)in Equation1does not involve the sampling size N,that is,the total num-ber of picked clones.The fluctuation probabilities, and confidence intervals,depend only on the values of the observed counts.To understand why,we must remember that Equation1governs the results of strictly duplicated experiments.Given N clones are sampled,the most likely tags to be picked up are, intuitively,those corresponding to cDNA,the abun-dance of which is of the order of1/N,or larger(ac-cording to Equation3,the probability of finding a given cDNA with1/N abundance while picking up N clones is0.63,see also Equation13).Choosing a sampling size therefore corresponds to targeting a given subset of genes,the level of expression of which allows their tags to occur at reasonable fre-quencies.As expected,more reliable inferences can be made on clones corresponding to larger absolute frequencies(i.e.,the ones more often picked up). For example(see Table1),a variation in counts from 1–3(threefold increase)is not indicative of a signifi-cant(P<0.05)increase,whereas a variation from 4–12is significant at P<0.05,and a variation from 7–21is significant at P<0.01.For a gene expressed at a given rate,increasing the sampling size N leads to higher tag counts,and allows more stringent sta-tistical inference to be made,for the same propor-tional variation.Most often in practice one wishes to compare digital Northerns or gene profiles that have been computed from the random picking of different numbers of clones,N1and N2.The mathematical problem is now to establish the probability for a given cDNA(e.g.,interleukin-2)to be picked up x times when the sampling size was N1and y times when the sampling size was N2.Equation1then becomes(see Methods):STATISTICAL ANALYSIS OF TRANSCRIPT PROFILESGENOME RESEARCHp͑y|x͒=ͩN2N1ͪy͑x+y͒!x!y!ͩ1+N2N1ͪ͑x+y+1͒(2)Whereas Equation1applied to the analysis of fluctuation in counts in strictly identical experi-ments,Equation2now applies to the analysis of counts in experiments only differing by the total number of clones randomly picked up.In practice, Equation2will be used to analyze experiments per-formed on two different libraries,using different sampling sizes.As for Equation1,small p(y|x)are expected to characterize the genes exhibiting regu-lated expression,the relative abundance of which is unlikely to be the same in the two libraries.Table1.Confidence Intervals in Function of the Value of xThe value of x(first column),one of the occurrence numbers.The intervals are given for the95%(2=0.05) and99%(2=0.01)confidence levels.Up to x=20,the exact boundaries,immediately outside the confi-dence interval(first significantly different values)are indicated.A star is used when none are possible.For larger values,the boundaries are given as percentages to be subtracted or added to x.Ricker’s confidence interval characterizes the value of,not y(see Methods).The use of a flat p()prior distribution results in the most stringent test,as expected.Although the number(N)of clones sampled does not appear in the expres-sion of p(y|x)(Equation1),its influence shows in the fact that the confidence interval becomes proportionally smaller as x(and y)increases(e.g.,1¨7has the same statistical significance as40¨60).For the same expression level,larger N will result in larger absolute values for x and y,making the detection of significant differential expression more sensitive.Comparison with Fisher’s(2×2)Exact TestThe(2ן2)contingency tables arising from treat-ment versus control experiments are traditionally analyzed with Fisher’s exact test(Siegel1956; Agresti1996).Differential EST count data can be presented in a tabulated form so as to suggest the use of this test,as follows:Brain cDNA library Liver cDNA libraryNumber of actin ESTs211 Number of other ESTs9981189Total clones sampled10001200 The statistical significance according to Fisher’s exact test for such a result is4.6%(two-tail P-value,i.e.,the probability for such a table to occur in the hypothesis that actin EST frequencies are in-dependent of the cDNA libraries).In comparison, the P-value computed from the cumulative form (Equation9,see Methods)of Equation2(i.e.,for the relative frequency of actin ESTs to be the same in both libraries,given that at least11cognate ESTs are observed in the liver library after two were observed in the brain library)is1.6%.Fisher’s(2ן2)exact test is always more conservative than our test(e.g., Fisher’s P-value of1.6%requires a2→13EST count transition in the above setting).Besides being too conservative,there is a more fundamental difficulty in using this test to analyze EST count data.The sampling scheme assumed by Fisher’s exact test in principle requires the total number of data values in the contingency table to be fixed,as well as both the row marginal total and the column marginal totals. In our prospective experimental situation,only the column marginals(i.e.,the numbers of clones sampled from each library)are fixed.The extension of Fisher’s exact test to cases where only one set of marginal totals is fixed(Tocher1950)is still contro-versial.In the context of the above EST counting results,there is an additional problem with the lack of homogeneity in the definition of the‘‘other EST’’category.This category represents different subsets of transcripts for different libraries.The use of Fisher’s(2ן2)exact test is more natural for a different type of EST data analysis:the study of library-dependent alternative transcripts of the same gene(i.e.,splice or polyadenylation vari-ants)(D.Gautheret,O.Poirot,F.Lopez,S.Audic, and J.-M.Claverie,in prep.).Here,the results for an hypothetical gene G1may look as follows:G1-relatedtranscripts inbrain libraryG1-relatedtranscripts inliver library Long-form mRNA210Short-form mRNA83Total G1-relatedclones1013where the alternative categories are unambiguously defined and refer to the same objects.For example, the above results constitute good evidence that G1is expressed in different forms in those tissues(Fisher’s exact test two-tail P-value=1.2%).False Leads in the Selection of Candidate GenesA crucial measure of the power of statistical signifi-cance tests is their rate of false alarm,that is,how often random fluctuations are expected to be mis-taken for significant differences in the results.When analyzing the transcript profiles from two different libraries,a false alarm would cause a gene to be deemed differentially transcribed,whereas in fact it is not.The rate of false alarm is therefore a direct estimate of the fraction of false leads,when search-ing for differentially expressed genes on the basis of differences in tag counts.The rates of false alarm associated with the P<0.01and P<0.05confi-dence intervals listed in Table1have been com-puted by Monte-Carlo simulation on the basis of two experimental sequence tag distributions(Table 2;Fig.1).The rate of false alarms associated with the use of Equation1(in fact,its cumulative form Equa-tion9,see Methods)is very small for genes repre-sented by small tag counts and slowly increases for higher tag counts,without ever exceeding the se-lected significance level.Such good behavior vali-dates the use of the confidence intervals(Table1) computed from Equation1and Equation9to assess the statistical significance of variations in digital Northern data.The curves labeled‘‘window’’char-acterize the very similar behavior of a slightly less conservative derivation of the same test(see Meth-ods,Equation15).For comparison,Figure1also presents the behavior of another test,based on an inappropriate application of Ricker’s confidence in-tervals(Ricker1937)(see Methods).DISCUSSIONAn appropriate statistical test is now at our disposal to begin analyzing digital gene expression profiles STATISTICAL ANALYSIS OF TRANSCRIPT PROFILESGENOME RESEARCHin a more quantitative way.For example,the test can be used to determine how many genes appear regulated at various confidence levels using the data from a typical experiment(e.g.,sampling a thou-sand clones).We analyzed the data gathered by Okubo et al.(1995)on the human promyelocytic leukemia cell line HL60induced by dimethylsulfox-ide(DMSO)or tetradecanoylphorbolacetate(TPA). Table3shows the21EST classes the occurrences of which exhibit significant variations at the1%level. Most of the corresponding genes make biological sense in term of differentiation along the granulo-cyte or monocyte pathways.This example serves to discuss a subtle point in the interpretation of the P values computed from Equation1,2,and9.Rigorously,these equations apply to the case where a given gene(e.g.,lipocor-tin)would have been selected for scrutiny before looking at the differences in cognate tag counts be-tween libraries.When comparing two libraries with-out specifying in advance the transcripts we want to follow,and then focusing a posteriori on any of those exhibiting significant variations,the average number of expected false positive N f a l s e is N false=PN species,where N species is the number of dif-ferent transcript species encountered and p is a given significance level.For instance,in the experi-ment analyzed in Table3,N species is of the order of 600(Okubo et al.1995).It is therefore possible that up to four(600ן7ן10מ3)out of the21transcript species listed in Table3are not truly differentially expressed.Therefore,when two libraries are compared without prior gene selection,the use of a predeter-mined significance threshold is not advisable.The P values computed from Equation1,2,and9should simply be used to rank all observed variations by order of decreasing statistical significance(analo-gous to how‘‘similarity hits’’are listed after data-base searches).The end-users can then make their own choice about the number of candidate target genes to be retained from the top of the list,bearing in mind the corresponding number of expected false positives.Although the present interpretation of a digital Northern focuses on the genes exhibiting the most spectacular differential expressions,there is already ample evidence that small changes can cause drasticTable2.Publicly Available Distributions of Sequence Tags(Left)Data from Velculescu et al.(1995):Frequency of occurrence of each of the428transcriptspecies represented in840SAGE tags randomly generated from a3Ј-directed cDNA library fromhuman pancreas.(Right)Data from Okubo et al.(1992):Frequency of occurrence of each of641transcript species represented in982randomly sequenced clones from a3Ј-directed cDNAlibrary from human liver cell line HepG2.AUDIC AND CLAVERIE990effects.Disease states caused by haploinsufficiency and trisomy suggest that2→1or2→3propor-tional changes in expression level may be of biologi-cal significance.Table1shows that there is no theo-retical limit to the detection of such small variations from the comparison of digital expression patterns. Simply,the sampling size has to be increased enough for the required numbers of cDNA tags to reach a significance threshold(for instance 40→60,for a confidence level of95%).Analog hybridization-based methods(Fodor et al.1991;Lennon and Lehrach1991;Gress et al. 1992;Southern et al.1992;Guo et al.1994;Matson et al.1995;Nguyen et al.1995;Schena et al.1995; Zhao et al.1995)are traditionally opposed to digital tag-counting methods(Okubo et al.1992;Matsub-ara and Okubo1994;Lee et al.1995;Okubo et al.1995;Vel-culescu et al.1995)for theanalysis of differential geneexpression.Both types ofmethods are sensitive to thequality of the original messen-ger RNA preparation and/orcDNA libraries.Analog meth-ods promise higher through-put,lower cost,and have thecapacity of studying transcriptson a much wider scale of abun-dance.They are therefore ex-pected to supersede digitalmethods.On the down side,however,hybridization signalsare not easily reproducible,andcan be affected by many un-known properties such as thecDNA library complexity,aswell as clone and sequence spe-cific features(e.g.,insert size,nucleotide composition,pres-ence of repeats,secondarystructure,triple helix interac-tion,etc.).Therefore,the hy-bridization-based methods re-quire an estimation of the dis-persion of the signal associatedwith each clone(i.e.,enoughrepetitions of each experi-ment),and multiple standard-ization and calibration proce-dures to allow the meaningfulcomparison of hybridizationpatterns obtained from varioussources(tissues,cell types,etc.) or from different membranes or chips.This is far from routine and has yet to be worked out.In con-trast,and thanks to the unique properties of the Poisson distribution,digital methods have the ca-pacity of providing a quantitative assessment of dif-ferential expression without the repetition or the standardization of individual tag-counting experi-ments.The statistical analysis presented here pro-vides an objective method to analyze digital tran-script profile data,and adapts it to fit(1)the num-ber of leads one wants to be followed;(2)the fraction of false clues to be tolerated;and(3)the level of modulation in gene expression considered of biological interest.A program is available on our web site(http:// rs-mrs.fr)to compute the confidenceFigure1Rate of false alarm computed according to the confidence intervalslisted in Table1.(Top)Monte-Carlo simulation of the random sampling of840tags distributed according to the data from Velculescu et al.(1995;see Table2).(Bottom)Monte-Carlo simulation of the random sampling of982ESTs distrib-uted according to the data from Okubo et al.(1992;see Table2).The fre-quency of false alarm was computed for two significance levels(2⑀=5%,leftand2⑀=1%,right)and plotted in function of the tag class size(from1–64forVelculescu et al.,from1–22for Okubo et al.).In all cases,the rate of false alarmincreases up to a plateau for larger class sizes.The test(cumulative form ofEquation1)derived from the flat p()prior shows perfect behavior with amaximal rate of false alarm always less than the significance levels(brokenlines).The test(cumulative form of Equation15)derived from the window p()prior exhibits a slightly higher rate of false alarms.Both versions of the testexhibit conservative behaviors for class size<5,with a false alarm rate even lessthan expected.In contrast,Ricker’s confidence intervals(Equation12)aregrossly inadequate and lead to false alarm rates up to four times the significancelevel.Graphs are computed from the analysis of1000repetitions of each ex-periment.STATISTICAL ANALYSIS OF TRANSCRIPT PROFILESGENOME RESEARCHintervals corresponding to arbitrary significance lev-els and sampling size N1and N2.METHODSLet us denote p(x)the probability to observe x sequence tags of the same gene(i.e.,from the3Јend of the same transcript) when N cDNA clones are picked randomly.For each transcript representing a small(i.e.,less than5%)fraction of the library and Nജ1000,p(x)will closely follow the Poisson distribu-tion:p͑x͒=e−xx!(3)whereis the actual(albeit unknown)number of transcript of this type per N clones in the library.If we duplicate this ex-periment(i.e.,once again randomly pick N clones of the same library and generate sequence tags),we will now observe y occurrences of the same transcript.What is the probability of the various y values?An approximate solution consists in us-ing x as the maximum likelihood estimate forand compute the probability for y occurrences given a Poisson distribution of mean=x:p͑y|x͒=e−x x yy!(4)Equation4is not symmetrical in x and y.This is an ob-vious flaw as the probability should not depend on which of the x or y values were observed first.p(y|x)=p(x|y)should hold provided that an equal number N of clones is sampled in both experiments.Equation4is not the correct formula,be-cause we have not yet taken into account the fluctuation of x around the unknown mean.To account for the fact that the actual value ofis unknown,we have to integrate Equation4 over all possiblevalues:p͑y|x͒=͐0ϱdp͑d=|x͒p͑y|d=͒(5) p(d=|x)in Equation5is the probability that the actualTable3.List of ESTs Exhibiting Significant(P<0.01)Differences inAbundance in the HL60Cell Line Induced by DMSO or TPAEST ID HL60HL60+TPA HL60+DMSO Significance418221013ן10מ7211241024ן10מ71982328ן10מ735616203ן10מ638012106ן10מ513541206ן10מ528514811ן10מ4201501102ן10מ424401143ן10מ429313613ן10מ429211015ן10מ465014525ן10מ433515339ן10מ444410412ן10מ316740814ן10מ31550834ן10מ38616107ן10מ33056207ן10מ318060607ן10מ318080607ן10מ317660607ן10מ3Only the probability(computed according to Equations7and8)corresponding to the most significanttransition(numbers in bold)is listed(Okubo et al.1995).The total EST numbers sampled from the HL60,HL60+TPA and HL60+DMSO cDNA libraries are845,845,and1058,respectively.ESTs418,211,356,285,293,292,650,335,444,861,305corresponding to ribosomal proteins,and EST380,a tag to an unkown gene,exhibit a marked reduction of expression level in the DMSO-and/or TPA-induced differentiated states.Inconstrast,ESTs135(ferritin),2015(LD78/macrophage inflammatory protein),1674(methionine adenosyl-transferase),155(thymosin-4),1806(lipocortin),1808(thymosin-10),and1766(a metallothionein)appear more abundant in the TPA-induced state,also highly enriched in EST19(the ubiquitous elongationfactor1-␣).-Actin(EST244),is the only markedly increased tag in the DMSO-induced state.EST numbers,abundance data,and protein assignments are from the‘‘body map’’public expression data repository athttp://www.imcb.osaka-u.ac.jp(K.Okubo and K.Matsubara).AUDIC AND CLAVERIE992abundance of a given transcript isgiven that x occurrences of a cognate tag have been observed in one experiment.The second term in the integral is the probability of drawing y occurrences given a Poisson distribution of mean:p͑y|d=͒=e−yy!(6)Using Bayes’theorem p(d=|x)can be written asp͑d=|x͒=p͑x|d=͒p͑d=͒͐0ϱdЈp͑x|d=Ј͒p͑d=Ј͒(7)To evaluate Equation7,we need to define the prior dis-tribution p(d=).The least constrained hypothesis(i.e.,with the least information content),is to attribute an equal a priori probability to allvalues in the[0,ϱ]range.Incorporating such a flat prior in Equation5leads top͑y|x͒=1x!y!͐0ϱde−2͑x+y͒(8)From the definition of the⌫function for integer arguments we observe that͐0ϱde−2͑x+y͒=͑x+y͒!2͑x+y+1͒and finally obtain the expression given in Results:p͑y|x͒=͑x+y͒!x!y!2͑x+y+1͒(1)This equation can be used in a wide variety of experi-mental situations.Equation1defines the probability of ob-serving x and y occurrences of the same rare event in dupli-cated experiments,regardless of the detailed probability dis-tribution of those events among the set of possible outcomes. In particular,in the context of transcription profiles,p(y|x) can be evaluated regardless of the distribution of each tran-script(provided it is rare)within a cDNA library.To compute the confidence intervals listed in Table1,we made use of the cumulative distributions:C͑yഛy min|x͒=͚y=0yഛy min p͑y|x͒(9a)D͑yജy max|x͒=͚y=y maxϱp͑y|x͒(9b) These equations allow the computation of an interval [y m i n,y m a x]⑀s u c h a s C(yഛy m i n|x)ഛ⑀a n d D(yജy max|x)ഛ⑀.Given that an event is observed x times in one experiment,the number y of occurrences of this event in a duplicate experiment is expected to fall within the interval [y min,y max]⑀with a probability of1–2⑀.Equation9,a and b, can therefore serve as a significance test when comparing,for instance,the results of sampling N clones from two different libraries.For2⑀small(e.g.,5%or less),y values falling outside the[y min,y max]⑀interval correspond to p(y|x)<<1,and point out significant differences between the two experiments. They should include differentially expressed genes,for ex-ample,for whichis different in the two libraries.Generalization to Different Sampling SizesWhen different numbers of clones N1and N2are sequenced from the same library,Equation5becomesp͑y|x͒=͐0ϱd2͐0ϱd1p͑d1=1|x͒p͑y|d2=2͒␦ͩ2מN2N11ͪ(10) where the two abundance values1and2are forced in the same ratio as N1and ing the same bayesian argument as before(Equation7)leads top͑y|x͒=1x!y!ͩN2N1ͪy͐0ϱd1e−1ͩ1+N2N1ͪ1͑x+y͒(11)the last integral is simply͑x+y͒!ͩ1+N2N1ͪ͑x+y+1͒leading to the formula presented in the Results section:p͑y|x͒=ͩN2N1ͪy͑x+y͒!x!y!ͩ1+N2N1ͪ͑x+y+1͒(2)Ricker’s Confidence IntervalThe confidence interval computed from Equation1(and its cumulative form,Equation9,a and b)is different from one introduced previously by Ricker(1937)although,at first,the two may appear to be related.Given x occurrences of a sequence tag,Ricker’s formula defines a confidence interval[min,max]x for(again the actual number of transcripts of this type per N clones in the library)such asp͑kഛx͒=͚k=0x e−maxmax kk!ഛ␣2(12a) andp͑kജx͒=͚k=xϱe−minmin kk!ഛ␣2(12b) where␣is typically5%or1%.Ricker’s confidence intervals for various values of x are given in Table1.Those intervals are close to those computed from Equation1,but delineate the range of likelyvalues,not y(the number of occurrences of the same event in a duplicated experiment).It is possible for x and y to fall outside each other’s Ricker’s confidence interval [min,max],while still being nonsignificant fluctuations around the samevalue.The confidence intervals computed from Equation12,a and b,are therefore too narrow to prop-erly define significant discrepancies between x and y.The false alarm rate associated with the use of Ricker’s confidence in-tervals is too high(Fig.1).However,an interesting use of Equation12,a and b,is the estimation of the range of possible frequencies[min,max]x=0for cDNAs not yet encountered after picking N clones.For example,the95%confidence interval is given by:0<N<3.7(13) That is,the abundance of a cDNA not picked up among STATISTICAL ANALYSIS OF TRANSCRIPT PROFILESGENOME RESEARCH。
